Pseudocolor in Pure and Applied Mathematics, a Free on-Line e-Book with
Source Code

Douglas Youvan

ISBN 978-0-615-43573-2

From the Youvan Foundation, a nonprofit 501(c)3 educational
charity in formation.

Pseudocolor is very useful in the representation of
computational output that involves complex data and phenomena - including
numerical tensors and digital images. Many of the examples in this e-book are pedagogical in
nature and useful for introducing students to higher levels of
Mathematics. Our full source code is published in Mathematica ".nb" and
Adobe ".PDF" formats; the latter has fully executed graphics embedded.
We have used a very limited
vocabulary of Mathematica functions so this code can be read easily and ported
to other languages. As such, the simple syntax of Mathematica can
serve as a flow chart for other implementation. Our 'pure' examples
include work in matrix algebra, fractals, tuples, permutations,
transcendental numbers, statistics, computation speed, and P=NP. Our
applied examples include work in biology, molecular biology, biotechnology,
chemical kinetics, energy conversion, thermodynamics, spectroscopy,
signal processing, image processing, digital imaging spectroscopy, graphics.

As a way of introducing the idea of using pseudocolor to represent
mathematical concepts and applications, three exemplary graphics are shown
below. These graphics come from the fields of aerodynamics (Savonius
rotor), biology (genetic code), and pure math (Tuples). Following these
examples, you will find a Forward and Table of Contents with hyperlinks to
explanations and source code. Currently, 16 examples are completed, 4 more
are in progress, and as many as 60 more examples are planned. Please
contact the author if you have any comments
- as this is a work in progress. Blue text indicates active hyperlinks.

With the advent of inexpensive
computers running with gigahertz speed and gigabyte RAM, it appears that
issues of compiled code and concise memory management might become concepts of
the past for most applications. Such statements always look funny in
hindsight, after computers become still faster and cheaper. That makes a
high-level language such as Mathematica even more attractive as a future
platform. A developer using a platform such as Mathematica can take
advantage of the work of a larger group, such as Wolfram Research, to free
themselves from monolithic operating systems. The developer is then
free to code and spend most of their time in logic rather than in ever-changing
fundamentals underlying developer studios. We also
anticipate that Mathematica code is easy to read - even without
flowcharts - and that it can be easily ported to another
language.

It is important to visualize mathematics in order to learn. For
example, calculus is more easily learned when it is combined with analytical
geometry. One sees derivatives as the slope of functions, and one can
picture the concept of an integral as simply the area under a functional curve.
Our sense of vision can not be neglected in the path to learning higher math.
That is why physicists are often the people that advance math - they
actually see what they are doing. A century ago, quantum
mechanics began to show us the picture of atoms in the form of
wave functions that graph as easily understood electron clouds.
Without this visualization, it would be very difficult to
appreciate the new, underlying mathematics. In electromagnetism
and optics, theorems are best visualized as vectors
pointing away from surfaces. Many of us have learned math
through physics.

Biotechnology is also increasing important in applied and pure
mathematics. One seeks to understand our 3 billion nucleotide
genome and the interaction of drugs in terms of combinatorial
possibilities. This opens up the entire field of discrete
mathematics and combinations to the applied mathematician for
use in biotech. Rapid prototyping of new algorithms in these
fields is very important. That is aided by high level
languages that can be used by an individual researcher or a sole
practitioner.

Images and source code from this e-book are also made available
to the public for any lawful use (worldwide) via our PD-Self
deposition at
Wikimedia.
As of 2008, the public domain
SAGE platform will run Mathematica computer code
and display, for example, using the Microsoft Internet Explorer.

In this second mathematics e-book, I change emphasis from
pseudocolor to discrete mathematics. For now, I will be publishing both
PDF and nb versions of the code; later, I will come back and write descriptions
of each program. From the title of each example and comment statements
within the code, you should be able to deduce my use of the code and possible
uses you might have. We begin work in Version 9 of Mathematica,
with an update to Version 10 expected soon. Publishing began May 22,
2014 from code already completed. I expect this book to take a few years
to finish.

Einstein and Lorentz Play Marbles. In this short story, I use
relativity to reduce an N! problem to N^2 both in computation time and scaling.
Einstein's time dilation equation for the Twin Paradox is identified as a
solution to P v NP.